Absolutely Summing Operators on non commutative $C^*$-algebras and applications

Abstract

Let $E$ be a Banach space that does not contain any copy of $\ell^1$ and $\A$ be a non commutative $C^*$-algebra. We prove that every absolutely summing operator from $\A$ into $E^*$ is compact, thus answering a question of Pe\l czynski. As application, we show that if $G$ is a compact metrizable abelian group and $\Lambda$ is a Riesz subset of its dual then every countably additive $\A^*$-valued measure with bounded variation and whose Fourier transform is supported by $\Lambda$ has relatively compact range. Extensions of the same result to symmetric spaces of measurable operators are also presented.